Potential symmetries and solutions by reduction of partial differential equations

Abstract

The authors determine some necessary conditions for a given partial differential equation E, written in conservative form to admit a potential symmetry (PS). A PS of E is a point symmetry of the auxiliary system Sp is obtained introducing a potential as further unknown function, then a PS leads to the construction of solutions via the classical reduction method. Given a PS, they introduce an algorithm that allows us to determine a class of E-solutions which includes the ones obtained as invariant solutions under the related point symmetry of Sp. As examples, they consider a Fokker-Planck equation, a wave equation in nonhomogeneous media and a quasilinear wave equation.